The Riemann Hypothesis

EasyChair Preprint no. 3708, version 49

5 pagesDate: May 11, 2021

Abstract

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. A theorem due to Erhard Schmidt implies that $S(x)$ changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities $\delta(x) \leq 0$ and $S(x) \geq 0$ are satisfied for some $x \geq 127$, then the Riemann Hypothesis should be false. However, the Mertens second theorem states that $\lim_{{x\to \infty }} \delta(x) = 0$. Moreover, a result from the Gr\"{o}nwall paper could be restated as $\lim_{{x\to \infty }} S(x) = 0$. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis.

Keyphrases: Divisor, inequality, number theory